- 15 Simultaneity in General Relativity and in Quantum Mechanics
- Johns Hopkins University Press
- book
- View Citation
- Additional Information

In contrast to the voluminous literature on the notion of distant simultaneity in the special theory of relativity, the study of this concept in the general theory of relativity, or only in noninertial space–time systems, has been given rather limited attention. One reason is that a serious study of this subject requires some knowledge of the tensor calculus and differential geometry that philosophers seldom possess. Another reason is the restricted applicability of this notion in the general theory. In fact, most treatises on general relativity ignore this topic completely.1 In 1907 Einstein himself recognized that the standard operational deﬁnition of this concept, as he proposed in his 1905 relativity paper, and hence also his deﬁnition of the concept of time (Edt in chapter 11), lose their apC H A P T E R F I F T E E N Simultaneity in General Relativity and in Quantum Mechanics 1 Exceptions, which will be used in our presentation, are L. Landau and E. Lifshitz, The Classical Theory of Fields (Cambridge, Massachusetts: Addison-Wesley, 1951, 1962); H. Arzeliès, Relativit é Généralisé–Gravitation (Paris: Gauthier-Villars, 1961); S. A. Basri, “Operational Foundations of Einstein’s General Theory of Relativity,” Reviews of Modern Physics 37, 288–315 (1965); J. W. Kummer and S. A. Basri, “Time in general relativity,” International Journal of Theoretical Physics 2, 255–265 (1969); I. Ciufolini and J. A. Wheeler, Gravitation and Inertia (Princeton, New Jersey: Princeton University Press, 1995). 272 Concepts of Simultaneity plicability in noninertial systems and, in particular, accelerated systems. For in part V of his comprehensive 1907 essay on relativity (listed in note 40 of chapter 7), from what later became known as the equivalence principle, Einstein derived the result that the velocity of light c ceases to be a universal constant and depends on the gravitational potential & in accordance with the equation c c(1 &/c2). (15.1) The mathematical apparatus Einstein used in these considerations did not yet include the possibility that the metric of the space–time under consideration can itself be a function of time. The possibility of a time-dependent metric seems to have occurred to Einstein not before 1914 or 1915 when he recognized the importance of formulating generally covariant equations for physical processes by means of the tensor calculus and Riemannian space– time geometry. At the end of chapter 7, it was pointed out that Einstein proposed that the coordinate system to be dealt with should be a reference system “in which the Newtonian equations hold.” It was also shown that within the context of his 1905 relativity paper such an assumption involved a vicious circle. Further , it was claimed that not every kind of reference system admits the standard deﬁnition of distant simultaneity. As we will now see, by using the Riemannian metric, it is possible to determine exactly what kind of coordinate system admits Einstein’s standard deﬁnition of distant simultaneity. To make the following considerations accessible to more than the mathematical expert we will simplify their treatment as much as possible. We must assume, however, that the reader knows that in the general theory the invariant separation ds between two inﬁnitesimally close events xa and xa dxa is given by ds2 g dx dx (, 0, 1, 2, 3), (15.2) where the coefﬁcients g are functions of the coordinates xa etc., x0 is the time coordinate, and the Einstein summation rule over repeated Greek indices is applied. The metric (or gravitational ﬁeld) is called stationary if the g values are time independent, that is, when g,0 0 (which is the case if it admits a timelike Killing vector ﬁeld). It is called static if it is stationary and, in addition, g0 0 for 1, 2, and 3 (which is the case if the timelike Killing vector ﬁeld is orthogonal to a foliation of spacelike hypersurfaces). To study the problem of whether it is possible to deﬁne simultaneity in general relativity analogously with its standard deﬁnition in special relativity we consider a clock UA at a location A and a clock UB at a location B in- ﬁnitesimally near to A. Let dx0 AB denote the coordinate time interval required by a light signal to travel from A to B and dx0 BA the coordinate time interval for the signal to travel from B to A. For a light signal ds2 0 so that equation (15.1) can be written in the form 0 g...

**You are not currently authenticated.**

**OR**Login